Random walks on co-compact fuchsian groups

Abstract

It is proved that the Green's function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence R. It is also shownthatAncona'sinequalitiesextendtoR,andthereforethattheMartinboundaryforR-potentials coincides with the natural geometric boundary S 1 , and that the Martin kernel is uniformly Holder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, p n (x;y) Cx;yR n n 3=2 .